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 Post subject: Dot Product and Euclidean NormPosted: Fri, 22 Oct 2010 02:37:49 UTC
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Joined: Fri, 17 Sep 2010 00:13:04 UTC
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Supposer that A is a real m x n matrix such that, for any vectors x and y in , if , the . Show that, if x and y are any two vectors in such that ||x||=||y||, then ||Ax||=||Ay||, where || || denotes the familiar Euclidean norm defined by the dot product. Also show, that if A is not the zero matrix, then .

Here it was I have so far:

Proving the above statement was part of the hint given in class, but I don't see how that can be used to prove ||Ax||=||Ay||. Also, for the second part I assume m has to be larger because we are in and multplying a matrix by a vector will result in n-1 dimensions, but is there more to that?

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 Post subject: Re: Dot Product and Euclidean NormPosted: Fri, 22 Oct 2010 09:38:39 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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soulrebel934 wrote:
Supposer that A is a real m x n matrix such that, for any vectors x and y in , if , the . Show that, if x and y are any two vectors in such that ||x||=||y||, then ||Ax||=||Ay||, where || || denotes the familiar Euclidean norm defined by the dot product. Also show, that if A is not the zero matrix, then .

Here it was I have so far:

Proving the above statement was part of the hint given in class, but I don't see how that can be used to prove ||Ax||=||Ay||. Also, for the second part I assume m has to be larger because we are in and multplying a matrix by a vector will result in n-1 dimensions, but is there more to that?

(1) Note that all your are actually .

(2) Look at the image of A.

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